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Einstein’s Static Universe:
An Idea Whose Time Has
Come Back?
Aubert Daigneault and Arturo Sangalli
A Tribute to Irving Ezra Segal (1918–1998)
W
hat is the shape of space? Is the universe finite or infinite? Did the world
have a beginning, or has it always
existed? These fundamental questions have intrigued and baffled humans since the most ancient times. But it was only
in the twentieth century, with the development of
powerful tools for probing the immensity of the
skies, that it became possible to explore the cosmos beyond our galaxy’s neighborhood.
Concurrently, advances in physics and mathematics provided the conceptual framework—the language, so to speak—in which to formulate comprehensive theories whose validity could be
objectively put to the test. Scientific answers to the
above questions finally appeared to be at hand.
Space-time is defined as the totality of pointlike
events—past, present, and future—in the history
of the world. A “pointlike event” means an “instantaneous event taking place at a point of space”
as opposed to an “event” that is extended in either
time or space, such as World War II. In 1917 Albert
Einstein proposed a model for space-time known
as the Einstein Universe (EU), in which the totality
of physical space is finite and curved [3]. “Nothing
in general relativity has intrigued the lay public
more than Einstein’s possibility of a closed, finite
Aubert Daigneault is professor of mathematics at the
Université de Montréal, Canada. His e-mail address is
[email protected].
Arturo Sangalli is professor of mathematics at Champlain Regional College, Lennoxville, Quebec, Canada. His
e-mail address is [email protected].
JANUARY 2001
spatial universe,” observed Theodore Frankel in
his 1979 introductory book to Einstein’s theory.
In EU, space may be mathematically described
as a three-sphere S 3 of fixed radius r , i.e., as the
boundary of a four-dimensional ball, by the equation u12 + u22 + u32 + u42 = r 2 . In Einstein’s model,
time had no beginning: it is infinite in both directions, and so the universe has always existed and
will always exist. Hence EU may be presented as
the cartesian product R × S 3 where R is the whole
real timeline.
In 1929 Edwin Hubble interpreted the galactic
redshift phenomenon detected by Vespo Slipher
starting in 1912—the change in the observed frequency of light waves—as a Doppler effect, that is,
as caused by the motion of a luminous source
away from the observer. The Doppler interpretation became known as the Expanding Universe
theory, whose most developed form is the cosmology of Friedmann and Lemaître and according
to which galaxies are moving away from each other.
As told in 1997 by David Dewhirst and Michael
Hoskin in The Cambridge Illustrated History of
Astronomy, “There is no doubt that the nearly
simultaneous detection of the redshifts and the
derivation of solutions of Einstein’s equations that
suggested that the universe would be expected to
expand greatly encouraged this interpretation.”
The Expanding Universe theory later begot the
Big Bang theory, which maintains, in addition to
universal expansion, that time and the universe
had a beginning, the “Big Bang”.
Hubble also stated his famous law: The
galaxies recede from each other with a velocity
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©2000 Index Stock Photography, Inc.
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Albert Einstein
proportional to the distance separating them. But
he did not rule out that, by virtue of some unknown
mechanism, the redshift might result from space
being curved. His reservations led him to write
in 1936 that “the assumption that redshifts are not
velocity shifts is more economical and less vulnerable, except for the fact that, at the moment,
no other satisfactory explanation [i.e., apart from
the Doppler effect] is known.” Once Hubble’s
expansion seemed to have been established as an
empirical fact, Einstein was forced to abandon his
static universe model.
The observation in 1964 of the so-called cosmic
background radiation (CBR) is often considered
to be the smoking gun supporting the assumption
that there was a Big Bang. But in fact the CBR
might have other causes. Erwin Finlay-Freundlich
and Max Born had already predicted the existence
of such radiation in 1953, eleven years before its
detection, on the basis of a stationary universe. The
interpretation of the CBR as a faint echo of the birth
of the universe gave greater credence to the Big
Bang theory, which became widely accepted as the
standard astronomical gospel.
A Two-Time Cosmology
By the early 1970s the rapidly increasing mass of
statistical data on quasars and galaxies provided a
substantial basis for questioning the Big Bang
cosmology. In 1972 Irving Ezra Segal, a professor
of mathematics at the Massachusetts Institute
of Technology, picked up where Einstein had left off
and proposed a variant of special relativity [13]:
chronometric cosmology (CC), so called
because it is based on the analysis of time. His 1976
book Mathematical Cosmology and Extragalactic
Astronomy [14] contains a detailed presentation of
the theory.
According to CC, Einstein’s model is the correct
one to understand the universe as a whole (i.e.,
global space-time), except that there are two kinds
of time: a cosmic or Einstein’s time t , and a local
or Minkowski’s time x0 , which is (perhaps!) the time
measured by existing techniques. In Segal’s words,
“The key point is that time and its conjugate
variable, energy, are fundamentally different in
the EU from the conventional time and energy in
the local flat Minkowski space M that approximates the EU at the point of observation.” Simply
put, Einstein’s cosmic time t is the “real” one,
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whereas Minkowski’s time is only an approximation of t .
Which coordinates, those of EU or those of M ,
are actually measured in observations is empirically
immaterial, as the two differ by unobservably
small amounts. Indeed, assuming “as is commonly
believed that r ≥ 108 lightyears,” the two “deviate
by less than one part in 1015 out to distances of
1 lightyear, or of less than 1 part in 106 out to
galactic distances. There is no apparent means to
detect such differences in classical observations”
[13, p. 766].
This two-time situation arises from the essentially uniquely defined conformal immersion of
Minkowski space M = R × R 3 , in which space is
ordinary three-dimensional Euclidean space, into
EU = R × S 3 . This immersion is unique up to the
group of causality-preserving motions of EU and
is a relativistic variant of stereographic projection. Minkowski space M can be thought of as
being tangent to EU just as the complex plane is
tangent to the Riemann sphere. In spite of this, the
immersion of M into EU preserves neither the time
coordinate nor the space coordinate in the factorizations of these space-times as “time × space”.
When units are chosen to make the speed of light
c equal to 1 , Einstein’s and Minkowski’s time
coordinates are related by the equation
(1)
x0 = 2r tan(t/2r ),
from which the relation
(2)
z = tan2 (t/2r )
of an observed redshift z to time of propagation t
or, equivalently, geodesic distance on the
3-sphere, may be derived essentially by simple
differentiation.
Equation (1) implies that as t varies from −π r
to +π r , ordinary time goes from minus infinity to
plus infinity. Hence eternity (past and future) in the
ordinary sense corresponds to a finite interval of
cosmic time, which cosmic time t , nevertheless,
varies over the whole real line. As for equation (2),
it reveals that for small values of t (or, equivalently, of the distance), the redshift varies as the
square of t , in contradiction with Hubble’s law,
which is linear. From (2) we also see that as r tends
to infinity, z tends to 0 . Hence, as envisaged as a
possibility by Hubble himself, the curvature of
space is the reason for the cosmic redshift in CC.
An application of l’Hospital’s rule to equation (1)
shows that, as expected, x0 tends to t as r tends
to infinity. The above-mentioned unobservably
small differences between the two times can easily be established from the series expansion of x0
in powers of t :
(3)
x0 ∼ t + t 3 /(12r 2 ) + t 5 /(120r 4 ) + . . . .
Because the radius r of the universe is very
large, it follows from (3) that only extragalactic
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observations can be relevant for telling the two
times apart. As Segal writes, “…in the absence of
precise observations of masses and distances outside the solar system, only the redshift and related
observations appear to fall in the category required
to distinguish, potentially at least, between the
two clocks” [10, p. 857].
Equation (2) may also be derived [12, 17] in (and
above) the observable frequency range by a rigorous analysis based on Maxwell’s equations governing light and, more generally, all electromagnetic
radiation. These equations are defined primarily
in M , but extend uniquely, along with their solutions of finite Minkowskian energy, to the larger
universe EU . It follows from this analysis that a free
photon will experience a redshift when propagated
over a very long period according to Einstein time.
In CC the Einstein universe EU is first thought of
as empty of matter just like Minkowski space-time
M , which is the ordinary space-time of special
relativity, and appears as a natural alternative to
the latter. In that sense EU belongs in the first place
to special relativity rather than to general relativity
(Einstein’s theory of gravitation) in whose framework it was presented by its initial proponent. EU is
uniquely arrived at in CC by means unrelated to
gravitation.
Thus a distinction is introduced in CC between
a space-time thought of as empty and a universe
which is inhabited with matter and which enjoys
only approximately the symmetries of the
underlying space-time. This distinction has no
counterpart in general relativity, where there is
no a priori geometry and where it is precisely the
introduction of matter that determines the
geometry.
Causality
Space-time defined as the totality of all events—
past, present and future—is, before anything else,
a partially ordered set; the relation p < q amongst
two events means that p precedes q . This relation,
known as the relation of causality, of precedence,
or of anteriority, is the most immediate observational data. It conceptually and psychologically
precedes the measurement of distances and
duration and is independent of any observer.
The models of space-time generally considered
plausible are known as Friedmann-RobertsonWalker (FRW) space-times. These FRW space-times
are all endowed with such a causality relation,
which derives from their pseudo-Riemannian
metrics, and they can all be immersed by essentially unique causality-preserving maps into the
Einstein universe, though the metric of time or
space is not preserved by such immersions. EU
thus appears as the maximal FRW space-time.
Such a space-time is defined as a product I × S
where I is an interval (which may or may not be
bounded from below or from above) on the real
JANUARY 2001
Photograph courtesy family of I. E. Segal.
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Irving Ezra
Segal
timeline and S , the space, is a three-dimensional
Riemannian manifold of constant curvature.
If dσ 2 denotes the metric of S , the FRW space-time
is endowed with the hyperbolic pseudo-Riemannian metric also known as the Lorentz metric
ds 2 = −c 2 dt 2 + r 2 (t)dσ 2 ,
where c is the speed of light and r (t) is a function
of time t in I known as the “radius of the universe
at time t ”. This is a mathematically unrestricted
scale factor; in particular, its time derivative r 0 (t)
may very well exceed the speed of light without
special relativity being violated. Indeed, in an FRW
space the distance between two stationary points
x and y of space increases at the rate r 0 (t) . The
popular inflation theory of Alan H. Guth [4], an
MIT physicist, claims that “…during the earliest
moments of time all space expanded far faster
than the speed of light.”
In Big Bang models, r (t) tends to 0 as t tends
towards the lower limit of I , the instant of the Big
Bang (excluded from I ), whereas in EU, r (t) is a
constant and I is the whole real line. Any such
FRW space-time is at each point (= event) locally
approximated by its tangent space, which is
essentially Minkowski space-time.
A vector x = (x0 , x1 , x2 , x3 ) in Minkowski space
M is called a time vector if x20 − x21 − x22 − x23 > 0 ;
it is said to be oriented towards the future if x0 > 0 .
The causality relation in M is defined by saying that
x < y if and only if y − x is a time vector oriented
towards the future. Special relativity’s essential
prescription is that the worldline of any material
point m , i.e., the set of all (t, x) such that m is at
spatial point x at time t , be a timelike curve, i.e.,
one whose tangent at any point is generated by a
time vector of the tangent Minkowski space at that
point of space-time.
A 1953 theorem of the Russian mathematicians
A. D. Alexandrov and V. V. Ovchinnikova, rediscovered a decade later by E. C. Zeeman, turned out to be
fundamental for CC (that was to be developed later)
as it entails that causality-preserving maps between
FRW spaces are the same as conformal maps. This
theorem is well expounded by Gregory L. Naber
in his 1992 book The Geometry of Minkowski
Spacetime.
In FRW spaces, in particular in EU, the causality relation is defined by declaring that an event
p precedes an event q if p can be joined to q
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by a directed curve such that at each point the
forward-looking tangent is a time vector oriented
towards the future.
Conservation of Energy
The Einstein universe and Minkowski space-time
are the only two space-times satisfying general
conditions embodying three fundamental physical
principles: first, the cosmological principle, also
known as the Copernican principle, which states
that there is no preferred direction or point in
space; second, the “principle of inertia”, the statement that there is no preferred timelike direction,
which implies the equivalence between observers
in relative motion at the same point; and third, the
statement that there is no preferred moment on
the time axis, which is a precondition for the
principle of conservation of energy to hold. Such
principles should be abandoned only in the face
of compelling evidence to the contrary. Segal contended that not a shred of such evidence exists and
that, on the contrary, the verifiable consequences
of these principles are confirmed by all available
astronomical data.
Segal’s explanation of the origin of the cosmological redshift is free from the adjustable parameters that intervene in the Expanding Universe
theory. Moreover, unlike the Big Bang cosmology,
CC does not need to resort to ad hoc scenarios—
such as “evolution” for quasars and radio sources,
and “inflation” at the hypothetical beginning of the
universe—reminiscent of the epicycles introduced
by ancient astronomers in their futile efforts to
reconcile the observed planetary orbits with a
geocentric universe. Guth’s still very fashionable
“inflationary universe” [4] was invented in the early
1980s to resolve two outstanding problems of
Big Bang theory, namely, the so-called “flatness”
and “horizon” problems, which simply do not arise
in CC.
A major merit of CC is to reestablish the
principle of global energy conservation, which
supporters of Big Bang cosmology are willing
to relinquish. “The explicit time dependence of
the Friedmann-Lemaître [expansionary] models
break Lorentz invariance as well as conservation
of energy-momentum. The loss of such theoretically fundamental as well as experimentally
well-established laws must be weighed in the
balance against the simplicity of a Doppler explanation for the redshift” [11, p. 4805]. Just as there
are two times in CC, there are correspondingly
two concepts of energy. The photon frequency
is a measure of the Minkowski energy, and the
redshift reduces that energy, which is therefore
not conserved. But in CC, thanks to its cosmic
time, this energy is not lost.
Freshly emitted photons have small spatial
support, and the energy of light composed of such
photons is conventionally inferred from laboratory
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measurements of the wavelength or the frequency,
in accordance with the Minkowski energy. However,
photons of extragalactic origin are necessarily
of cosmic spatial support when observed. And “a
direct measurement of the Einstein energy of an
extragalactic photon may depend on more than the
pointwise oscillation of the photon wave. Indeed
mathematical analysis shows that the Einstein
energy of a photon depends not only on the
oscillation frequency of the photon but also on the
number of oscillations, which appears beyond
the scope of the present methods to observe” [17,
p. 316].
As Roger Penrose says, “The utility of the
concept of energy, in general, arises from the fact
that it is conserved.” In Einstein’s theory of
general relativity, as opposed to what is the case
in the special theory of relativity, there is only a
differential (or infinitesimal) conservation law
and not an integral conservation law.1 The former
implies the latter only in the presence of a group
of temporal translations that are isometries of the
space-time model. Energy is integrally conserved
if the flux of energy through the boundary of any
compact 4-dimensional domain of space-time
vanishes. An example of such a compact domain
is a finite portion of space during a finite interval
of time [5, pp. 61, 62].
Aside from Minkowski space-time, the Einstein
universe is the only one amongst the FRW spaces
that has such a temporal group of isometries and
consequently the only one in which the principle
of integral conservation of energy is valid.
The energy that is conserved is, in particular,
that of electromagnetic radiation in vacuum “which
is basically all that is observable in large scale
astronomy.” Both electromagnetic energies, the
Minkowskian and the Einsteinian, are conserved in
their respective cosmos M and EU. “They are both
representable as the integral over space in the
respective cosmos of the square of the electromagnetic field, in terms of spatially natural
coordinates,…and the latter integral is always
larger” [17, p. 317]. The free electromagnetic field
can be construed as a Hamiltonian system in
infinitely many dimensions whose total energy is
as just said, and the dynamical group is induced
by the isometry group of temporal translations of
space-time, both in M and in EU.
Challenging Hubble’s Law
In 1929 Edwin Hubble made the sensational
announcement that the galaxies were receding
from each other. This phenomenon had already
been detected more than a decade earlier by Vespo
Slipher, an astronomer at the Lowell Observatory,
who had reported that at least some of the faint
1 We purposely refrain from using the word “local” in this
connection, as it sometimes means “differential” and
sometimes “integral”.
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patches of light then called nebulae appeared to
be receding from the Milky Way. When Hubble
plotted the observed redshift (or rather, in his
interpretation, the recession velocity v ) against the
estimated distance d for a small number of galaxies, he claimed to have detected a “roughly linear”
pattern, corresponding to the relation v = Hd ,
where the constant of proportionality H is known
as Hubble’s constant.
As Hubble himself admitted, the estimated
distances from which his law was derived were in
part uncertain, but the best then available. In the
ensuing decades other astronomers followed up
with their own verifications, and the linear redshiftdistance relation is by now generally accepted.
But Segal denounced the Hubble school for having based its confirmation of the law on observations of a relatively small class of objects, the bright
cluster galaxies, and without specifying a selection
criterion: “The central source list for such galaxies
has been the catalog of Abell, who stated: ‘In determining whether a cluster meets this criterion [for
inclusion in the catalog], it was assumed that the
redshifts of clusters are proportional to their distances.’ […] For this [circular argument] and other
reasons, the observational case for the Hubble Law
on the basis of cluster galaxy samples is at best
inconclusive” [16, p. 295].
Since the proposal of CC (chronometric cosmology), Segal and his collaborators, notably J. F.
Nicoll of the Institute for Defense Analyses, have
carried out an extensive analysis of all published
and reliable astronomical data on galaxies and
quasars in accordance with modern statistical
principles and in a reproducible manner. Their
conclusions have invariably been consistent with
the predictions of CC and at variance with those
of the expansionary theory and Hubble’s law. As
Segal points out, “Older statistical methods were
dependent on parametrizations that developed
from plausibility and convenience more than
direct observation. Rapid computer simulations
now make possible efficient nonparametric
methods that provide objective probabilities
of deviations of specified observations from
theoretical prediction, in circumstances far more
complex and realistic than would be possible by
theoretical estimation of probabilities.”
Using a sophisticated bootstrap statistical
technique that they call ROBUST, which takes into
account the so-called “observational cutoff bias”
making faraway celestial objects less likely to be
observed than closer ones, Segal and Nicoll demonstrated that the quadratic redshift-distance law
predicted by CC fits the available experimental
data on the more than 2,000 quasars of a reputed
comprehensive 1989 catalogue much better than
does Hubble’s law [16].
Because Hubble’s law is incompatible with
chronometric cosmology, its status has become a
JANUARY 2001
crucial test for this theory. In 1992 the statistician
Bradley Efron, a creator of the bootstrap technique, and Vahé Petrosian, a physicist, both at
Stanford University, published a paper [2] which
appears to disprove Segal’s claims. However, the
data on which they based their conclusions are
not readily available and have been “criticized
for being too inaccurate,” as Petrosian himself
acknowledges. In addition, the criteria for selecting a subsample of 492 redshifts used to confirm
the Hubble law was not specified.
In a 1997 article [6] Daniel Koranyi of the
Harvard-Smithsonian Center for Astrophysics
and Michael Strauss of Princeton University claim
that since its original announcement by Hubble,
“observational evidence has mounted steadily” in
favor of the linear law. This opinion conflicts with
Segal’s: “Careful investigation shows that the reported confirmation of Hubble’s Law has been
based on large components of wishful thinking….”
At any rate, discussing an earlier paper [15] by Segal
and his collaborators, the authors point to what
they perceive as a methodological pitfall affecting
Segal’s analysis against Hubble’s law: the sensitivity
of the estimation of the luminosity function (i.e.,
the probability distribution of the intrinsic luminosity of galaxies) to the number of bins into which
the sample used for that purpose is decomposed.
This, they contend, “could lead one to mistakenly
favor the quadratic law.” However, Koranyi and
Strauss’s statistics suffer from other shortcomings, some already pointed out by Segal in [15].
Such challenges to Segal’s work are rare. Over
the years there has been very little response to his
claims, and he has refuted all published criticism
of CC.
The Three Empirical Pillars of Big Bang
Theory
The tenants of the Big Bang theory interpret the
redshift as a Doppler effect indicating the expansion of the universe. They interpret the cosmic
background radiation as an echo of the initial explosion. The third empirical pillar of the theory is
called “Big Bang nucleosynthesis”, which claims to
predict the abundances of the four light elements:
deuterium, helium-3, helium-4, and lithium-7.
The much-trumpeted alleged concordance of
Big Bang predictions with observations goes essentially as follows. What is called the abundance
of a chemical element is the ratio of the amount
of this element in the universe to that of hydrogen, which is the most prevalent. In his 1989 review of Big Bang nucleosynthesis, Gary Steigman
of Ohio State University [18] explains that the only
parameter on which the predicted abundances depend is η , the universal ratio of nucleons to photons. For a narrow interval of values of η tightly
constrained by the observed abundances, “the
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standard hot Big Bang model predicts the individual abundances to lie precisely in this range.”
But, as Steigman says, “Abundances are not observed. Abundances are derived from the observational data, often following a long and tortuous
path involving theory.…Errors (or uncertainties),
often systematic, may be introduced at many steps
in the overall process of deriving abundances
from observational data. Furthermore we are here
concerned with primordial abundances. Even if
present day universal abundances were known to
arbitrary accuracy (which they are not!), we still
would have to employ theory and observation to
extrapolate back to obtain primordial (or at least
pregalactic) abundances. Additional errors (uncertainties) are surely introduced here too.”
N. Yu Gnedin and Jeremiah Ostriker of Princeton University, who are also Big Bang supporters,
declared in 1992: “Light element nucleosynthesis
has been a central pillar supporting the standard
FRW hot Big Bang cosmological model.…But there
are several confusing and apparently inconsistent
elements in the canonical picture which have led
to ‘patches’ which are quite ad hoc and are accepted
only because of our familiarity with them and our
basic belief that the underlying standard model is
accurate.”
Steigman still expresses some reservations in
a 1996 paper: “Indeed, the success of standard
Big Bang nucleosynthesis in predicting the abundances of the light nuclides with only one
adjustable parameter η restricted to a narrow
range while the abundances range over some 9
orders of magnitude is impressive indeed. However,
as with the nineteenth century standard model
[Newtonian gravity], some clouds have now
emerged on the horizon. Recent analyses…point
to a crisis unless the data are in error, or the extrapolation of the data are in error, or there is new
physics.” In more recent articles valiant efforts
are made to alleviate the crisis, which nevertheless
perdures.
These current difficulties of Big Bang cosmology
may ultimately lead to an “agonizing reappraisal”
of its basic tenet, the Doppler interpretation of
the redshift. But there is no sign of this yet, as the
notion of an expanding universe is by now such an
entrenched belief in the scientific community and
the population at large.
In any case, chronometric cosmology has responses to these contentions often presented as
definite “proofs” of the Big Bang theory. In CC the
cosmic background radiation is interpreted as the
equilibrium state of free photons that have been
scattered, reemitted, or absorbed many times in
the course of possibly thousands of circuits of the
3-sphere. In Segal’s words: “The observed blackbody form of the cosmic microwave background
is simply the most likely disposition of remnants
of light on a purely random basis, assuming the
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classic principle of the conservation of energy,
and is not at all uniquely indicative of a big bang.”
Back in 1926 A. E. Eddington had already calculated that the “temperature of space” produced
by the radiation of starlight would be found to be
3 degrees Kelvin, as is the case of the CBR.
In a private communication to one of the authors, Bertram Kostant, a mathematician at MIT,
summed up some recent joint work of his and
Nolan Wallach’s of the University of California at
San Diego as follows: “One of the major props for
the Big Bang theory is the black body radiation profile of the background radiation. Cosmologists say
that only an enormous heat source could account
for that profile. One of the things that we have
shown is that there exist solutions of Maxwell’s
equations in the Segal universe which have that
profile and have been around for all times. In
particular they did not require a heat source to
exist. In fact we have classified all such solutions.”
According to Segal and Zhengfang Zhou, a
mathematician at Michigan State University, the
treatment of nucleosynthesis is similar in CC and
in Big Bang theory to first order. “The difference
is only that a stochastic sequence of mini Bangs,
associated with, e.g., the formation of galaxy
clusters, replaces the unique Big Bang. Cluster
formation would be expected to be accompanied
by extremely high temperatures, which, as in
Big Bang theory, would be productive of light
elements” [17, p. 323].
The discovery in 1999 of apparently abundant
molecular hydrogen in the universe, a possibility
already envisaged by physicist Paul Marmet in
1989, may complicate matters even further for
Big Bang nucleosynthesis while at the same time
rendering unnecessary the concept of (to this day
undetected) nonbaryonic dark matter except for
maintaining Big Bang theory.
Which Model for the Universe?
Can Segal be right against almost the entire astronomical community? Isn’t there an experiment
that would settle the question? The problem is
that a direct, empirical confirmation of CC—or of
Hubble’s law for that matter—is not feasible
given the current technological means, the
greatest difficulty being to determine the distances
to astronomical objects. These distances are not
observed but are estimated in a complex manner
that depends on the model chosen. What can
be observed, apart from the redshift, are the
apparent brightness and the angular diameter.
Consequently, any comparison of the merits of
conflicting cosmological theories must hinge
not on direct observation but on a statistical
analysis of experimental data which would hardly
be tractable without high-speed computers.
Irving Ezra Segal died suddenly on August 30,
1998, a few days before his eightieth birthday. He
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was still actively engaged in research, pursuing,
amongst other mathematical endeavors, his lifetime
passion: to understand the universe. A 1940 Yale
graduate, he taught at Harvard, Princeton, and
Chicago before becoming an MIT professor in 1960.
He is the author of several monographs and countless scientific articles in prestigious journals and
was a member of the American and the Royal Danish Academies of Science. The width and depth of
his knowledge of physics and mathematics,
and of the interaction between the two, was
truly astonishing. His work may be summarized
as the search for the right mathematical models to
describe certain physical theories, such as cosmology and quantum field theory.
“Why has this work not received an adequate
evaluation?” asks Edward Nelson, a Princeton
University mathematician and former Ph.D. student of Segal’s, in a memorial article [1]. And,
answering his own question: “Part of the reason
lies in Segal’s style of scientific exchange—at times
it resembles that of Giordano Bruno (later burned
at the stake), who very shortly after his arrival
in Geneva issued a pamphlet on Twenty Errors
Committed by Professor De la Faye in a Single
Lesson. But part of the fault lies with cosmologists
and particle physicists intent on defending turf. The
time for polemics is past. Segal’s work on the
Einstein universe as the arena for cosmology
and particle physics is a vast unfinished edifice,
constructed with a handful of collaborators. It is
rare for a mathematician to produce a life work
that at the time can be fully and confidently
evaluated by no one, but the full impact of the
work of Irving Ezra Segal will become known only
to future generations.”
Segal never let up in his crusade against the
expansionary theory, alternating sound scientific
arguments with emotional tirades. In a 1992
letter to The New York Times, he reacted to the
interpretation of detected “ripples” in the CBR as
support for the Big Bang: “In this day of budding
mass movements that appear dangerously irrational, it is deplorable that some scientists, as
well as the media, indirectly support detachment
from rationality by effectively (if with the best of
intentions) catering to the popular fondness for the
Big Bang.”
A major objection often put forward against EU
as a realistic model of the universe, aside from its
alleged inability to explain the observed redshift,
is its purported and much decried instability: if EU
were somewhat perturbed, it would either expand
or collapse. However, the proof of such instability
is based on the widespread and rather naive
notion that the energy-momentum tensor of
Einstein’s gravitational equation can realistically
be represented by that of a perfect fluid, a notion
that is only a natural starting point and that
JANUARY 2001
ignores, in particular, any possible electromagnetic part in the tensor.
Segal expressed considerable skepticism [14, p.
189]: “…Probably still less justified physically is the
application of general relativistic hydrodynamics
to extragalactic questions such as the mass density and the stability of the entire Cosmos. The approximation of the distribution of galaxies by
a fluid is quite uncontrolled and open-ended;
at best, conclusions drawn in this way are merely
suggestive.”
Nobel laureate Hannes Alfvén, the founder of
modern plasma physics, i.e., the physics of electrically conducting gases, advocates what he calls
a “plasma universe”. Eric Lerner has championed
and popularized this idea in The Big Bang Never
Happened: “The plasma universe…is formed and
controlled by electricity and magnetism, not just
gravitation—it is, in fact, incomprehensible without electrical currents and magnetic fields.”
“Hannes Alfvén and his colleagues have shown
that a nonexpanding universe can be stabilized
by electric and magnetic forces,” writes Gerald S.
Hawkins of the Harvard-Smithsonian Observatories.
Already in 1920 Hermann Weyl had speculated
on the consequences of a spatially closed universe,
observing that if such were the case, then “it would
become possible for an observer to see several
pictures of one and the same star. These depict the
star at epochs separated by enormous intervals of
time (during which light travels once entirely round
the world).”
In the December 1998 issue of this journal,
Neil J. Cornish, a research fellow at the University
of Cambridge, UK, and Jeffrey R. Weeks, a geometer working as an independent scholar, took up
Weyl’s idea in the context of the Big Bang theory:
“When we look out into the night sky, we may be
seeing multiple images of the same finite set of
galaxies.” They even went one big step further
and, referring to an artist’s impression drawn on
the cover of that issue of the Notices, they wrote:
“It is possible that one of the galaxies seen in
this image is our own Milky Way, and the light we
receive from it has made a complete trip around
the universe.”
This possibility can be contemplated in CC as
well: “The transparency of cosmic space implies that
photons in EU will typically make many circuits of
S 3 before being absorbed or undergoing interaction. A free photon will be infinitely redshifted at
the antipode of S 3 to its point P of emission, but
on returning to P it will be in its original state, as
a consequence of the periodicity of free photon
wave function in EU” [17, p. 324].
We believe that Segal’s chronometric model for
cosmology merits serious consideration. It is true
that Albert Einstein, the most famous advocate of
a static, unevolving universe, later changed his
mind when he disavowed the “cosmological
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constant” Λ (lambda), a kind of antigravity factor
that he had introduced in his equations, just as
C. Neumann and H. Seelinger had done in a Newtonian context at the end of the nineteenth century, to prevent the universe from collapsing under
gravity [9]. Einstein is said to have referred to the
introduction of lambda as his “greatest blunder,”
and yet...
General relativity reduces gravitation to curvature. Einstein’s original equation implies that space
is flat, in the absence of matter, whereas
the modified equation that includes the Λ -term
allows empty space to have a curvature of its own
in addition to that resulting from the presence of
matter. The modified equation is the most general
obeying some minimal conditions and which allows
empty space to be curved or flat [8].
Saying that Λ is necessarily zero in the absence
of matter can be compared to saying that if it were
not for mountains, rivers, and valleys which define
local curvature on a piece of land, the earth would
be flat. The curvature defined by local topography
masks the global curvature of our planet, but, of
course, it is the other way around if one looks at
it from a sufficiently large distance.
“Theorists are already scrambling to understand
what 20 years ago would have been unthinkable:
a cosmological constant greater than zero yet much
smaller than current quantum theories predict,” observes Case Western Reserve University physicist
Lawrence M. Krauss in recent articles [7, for instance]. It is not unthinkable that a rehabilitation
of lambda, originally related to the radius r of S 3
by the equation r = Λ−1/2 , could give Einstein’s
static universe a second chance—and help Irving
Ezra Segal’s chronometric cosmology get the
attention it deserves.
After demonstrating the well-known Poincaré
theorem of Hamiltonian mechanics, the mathematician V. I. Arnold writes in his excellent treatise
Mathematical Methods of Classical Mechanics: “The
following prediction is a paradoxical conclusion
from the theorems of Poincaré and Liouville:
if you open a partition separating a chamber
containing gas and a chamber with a vacuum, then
after a while the gas molecules will again collect in
the first chamber. The resolution of the paradox
lies in the fact that ‘a while’ may be longer than the
duration of the solar system’s existence.”
Hence we may conclude on a rather speculative
but perhaps conciliatory note. Assuming CC, if
the concept of the totality of the matter dispersed
in space S 3 as a “gas of galaxies”, i.e., a gas the
molecules of which are the galaxies, is valid, as is
generally taught, then this gas is almost always in
equilibrium. However, if Poincaré’s theorem is
applicable to it, on rare scattered occasions, yet infinitely many times in past cosmic eternity, it has
taken very implausible configurations giving rise
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to fairly big “bangs”, and the same would be in
stock for the future.
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VOLUME 48, NUMBER 1